Rippon and Stallard [1]} showed that every component of
\(\mathcal {A}(f)\)  is unbounded. While a conjecture of Eremenko [2]}
saying that every component of \(\mathcal {I}(f)\)  is unbounded was recently
disproved [3]}, the result by Rippon and Stallard
implies of course that \(\mathcal {I}(f)\)  has at
least one unbounded component. In particular,
\(\dim \mathcal {I}(f)\ge \dim \mathcal {A}(f)\ge 1\) .
On the other hand, it was shown in [4]} that for all
\(\rho \in [1,2]\)  there exists a transcendental entire function \(f\)  such that
\(\dim \mathcal {I}(f)=\rho \) .
The main contribution of [4]} is the case \(\rho =1\) . The case
\(\rho \in (1,2)\)  is deduced from results in [6]} and [7]}.
Inspection of the proofs in [7]} shows that we actually may achieve
\(\dim \mathcal {A}(f)=\rho \)  for given \(\rho \in (1,2)\) .
For \(\rho =2\)  such an example is given by the exponential function;
this follows from the arguments in [9]}.
Further examples are given in [10]}.
Altogether we see that for all \(\rho \in [1,2]\)  there exists a transcendental
entire function \(f\)  such that \(\dim \mathcal {A}(f)=\rho \) .
